Convergence of $f_n(x)=nx$

Question

Consider the sequence of functions $$f_n(x)=nx$$ As $n$ gets larger, so does the gradient of the line passing through the origin. Graphically, as $n$ goes to infinity, this will converge to the vertical straight line $x=0$, which is not defined since the same value is mapped to (infinitely) many values.

Does this sequence converge to this vertical line? If so, how would one prove it, since you cannot just right this function limit as "$f(x)=...$"?

Answer 1

We cannot speak of convergence of functions if we don't know what the domain of the functions is. In your case:

1. If the domain of the functions is $\{0\}$, then the functions converge to the zero function.
2. On any set containing a nonzero element, the series of functions does not converge.